DESIGN OF A FUZZY PID CONTROLLER FOR APPLICATION IN SATELLITE ATTITUDE CONTROL SYSTEM. Willer Gomes. Evandro Marconi Rocco

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1 DESIGN OF A FUZZY PID CONTROLLER FOR APPLICATION IN SATELLITE ATTITUDE CONTROL SYSTE Willer Gomes National Institute for Space Research INPE ZIP: , São José dos Campos, SP, Brail willer.gomes@ahoo.com.br Evandro arconi Rocco National Institute for Space Research INPE ZIP: , São José dos Campos, SP, Brail evandro@dem.inpe.br Abstract: In this paper, a fu proportional-integral-derivative (PID) controller is designed to be applied in the satellite attitude control sstem equipped with reaction wheels. An analsis of the scaling factor is developed to verif its influence during an attitude maneuver. The settling time and the pointing accurac are the mission requirements that control sstem have to satisf. The equations and theoretical concepts are introduced throughout this paper. The results showed that the adequate selection of the scaling factor contributes to the improvement of control results. Kewords: Fu PID controller, attitude control sstem. 1 Introduction The attitude control sstem determines and controls the orientation of a vehicle in space. The information provided b attitude sensors is processed and filtered b an estimator state. Hence, this signal is compared with a reference. The error between the estimated state and the reference will be used b the control algorithm to activate the actuators appropriatel in order to reduce or eliminate this error. Several control techniques can be used in the controller. Well-known proportional-integral-derivative (PID) controller is the most widel used in industrial application because of its simple structure. However, control techniques which based on fu logic and modified PID controllers are alternatives to conventional control methods. Control strategies such as PID controllers are epressed in mathematical functions. Fu logic control, on the other hand, makes use of human knowledge and eperience to perform the control in a scheme similar to the human strateg, thus realiing an intelligent control. The design of a fu logic controller involves the construction of control rules and the parameter tuning. This paper shows the stud of the parameter tuning of a fu PID controller of a satellite. The modeling and simulation of an attitude maneuver of a satellite equipped with reaction wheels and fu PID controller was developed. In order to analsis the behavior of the fu PID controller response, different values of the scaling factor parameter of membership functions that define the fu sets have been investigated. The paper is organied as follow. In Section 2, mathematical model of the attitude movement of a satellite are presented as well as the reaction wheel models. Section 3 shows the general structure of a fu PID controller. Section 4 discusses the simulation results. Finall Section 5 provides conclusions. 2 Concepts and equations of the satellite attitude movement Different reference ais frames are chosen for different spacecraft tasks. In this paper, we adopted the orbit reference frame (X R, Y R, Z R ) as the reference coordinate frame in which the satellite is to be three-ais attitudestabilied. The origin of this frame moves with the center of mass of the satellite in the orbit. The Z R ais points toward the center of mass of the earth. The X R ais is in the plane of the orbit, perpendicular to the Z R ais, in the direction of the velocit of the spacecraft. The Y R ais is normal to the local plane of the orbit, and completes a three-ais right-hand orthogonal sstem. The satellite s ais frame is defined b X B, Y B and Z B and has its origin at the center of mass of the spacecraft. This frame is used as orientation for the attitude control and determination sstem. The inertial ais frame (X I, Y I, Z I ) has its origin at the center of mass of the earth. The spacecraft attitude can be represented b Euler ais and principal angle, Euler angles, Euler smmetric parameters (quaternion), Rodrigues parameters (Gibbs vector) and modified Rodrigues parameters (Tewari, Workshop on Space Engineering and Technolog, June

2 2007). The quaternions were used in the computations to avoid singularities in the results because their use do not require trigonometric calculations. However, representation b Euler angles has more phsical sense and serves to present the graphical results in this paper. Equation (1) is the required kinematic relationship between the Euler angles, φ, θ, ψ, and the angular velocit (ω) in the form of three, coupled, nonlinear first-order ordinar differential equations for rotations sequence. & φ & 1 θ = 0 ψ& 0 sφ sθ cφ sφ cφ ω sψ ω0 sφ ω + cψ cφ ω sθ sψ (1) where cβ cosβ and sβ sinβ, and ω 0 is the orbital angular velocit. A quarternion is a special set composed of four mutuall dependent scalar parameters, q 1, q 2, q 3, q 4, such that the three form a vector, called the vector part, and the fourth, q 4, represents the scalar part. The kinematic equation represented b quarternion is shown b Eq. (2). 0 ω ω + ω0 ω 1 q ω 0 ω ω + ω0 & = q (2) 2ω 0 ω0 ω ω ω ω ω0 ω 0 The instantaneous attitude depends not onl upon the rotational kinematics, but also on rotational dnamics which determine how the attitude parameters change with time for a specified angular velocit. Assuming that X B, Y B, Z B are the principal aes of inertia, the Eq. (3) represents the three scalar Euler s moment equations (Sidi, 1997). = I = I = I & ω + ω ω & ω + ω ω & ω + ω ω ( I I ) ( I I ) ( I I ) where I, I, I are the moments of inertia, and,, are the components of the total eternal moment vector acting on the bod, which is equal to the inertial momentum change of the sstem. The kinematic equations, along with Euler s equations of rotational dnamics, complete the set of differential equations needed to describe the changing attitude of a rigid bod under the influence of a time-varing torque vector (). omentum echange devices are actuators that allow changing the distribution of momentum inside the satellite, without altering the total inertial momentum of the entire sstem. The device consists of an electrical motor on whose ais is assembled a flwheel designed to increase its angular momentum of inertia and so deliver momentum to the satellite s bod as part of the attitude control sstem (ACS). The reaction wheel is a kind of momentum echange device and is used primaril to provide the satellite with sufficient torque for various attitude maneuvering tasks (Sidi, 1997). Soua (1987) presents the design and stud of an eperimental model of a reaction wheel. In this paper, we adopted the reaction wheel model suggested b Soua (1981), whose block diagram is shown in Fig. (1). (3) Figure 1. Block diagram of the reaction wheel. Equation (4) shows the transfer function between the torque (output) and the voltage (input) and Eq. (5) shows the transfer function between the angular velocit of the reaction wheel and the torque provided. Workshop on Space Engineering and Technolog, June

3 V r KvTvs = (4) 1 ( + T s) v Ω R 1 r = I R s (5) where r is torque provided b reaction wheel, V is the voltage input applied at wheel motor, K v is the wheel gain, T v is the time constant, Ω R is angular velocit of the reaction wheel and I R is its moments of inertia. The wheel parameters (K v and T v ) are given b Eq. (6) and (7), respectivel. R ma K v = (6) V R ma I RΩ R ma Tv = (7) Rma where Ω Rma is the maimum angular velocit; Rma is the maimum torque and V Rma is the maimum input voltage. 3 Fu logic control Fu logic controllers are as functional as the more conventional controllers, but the manage comple control problems through heuristics and mathematical models provided b fu logic, rather than via mathematical models provided b differential equations. This approach is useful for controlling sstems whose mathematical models are nonlinear or for which standard mathematical models are simpl not available. Creating machines to emulate human knowledge in control gives us a new wa to design controllers for comple plants whose mathematical models are not eas to specif (Nguen et al., 2003). Figure (2) shows the basic configuration of a fu logic control (FLC), which comprises four principal components: a fuification interface, a knowledge base (KB), decision-making logic, and a defuification interface (Lee, 1990). Figure 2. Basic configuration of fu logic controller (FLC). The fuification interface measures the values of input variables, performs a scale mapping that transfers the range of values of input variables into corresponding universe of discourse and performs the function of fuification that converts input data into suitable linguistic values which ma be viewed as labels of fu sets. The knowledge base comprises the knowledge of the application domain and the attendant control goals. It consists of a data base and a linguistic (fu) control rule base. The decision-making logic is the kernel of an FLC; it has the capabilit of simulating human decision-making based on fu concepts and of inferring fu control actions emploing fu implication and the rules of inference in fu logic. The defuification interface performs the following functions: a scale mapping, which converts the range of values of output variables into corresponding universe of discourse; and defuification, which ields a nonfu control action from an inferred fu control action (Lee, 1990). The fu PID controller used in this paper has two inputs and one output. These are error, change of error and control signal, respectivel. The overall structure of used controller is shown in Fig. (3). Workshop on Space Engineering and Technolog, June

4 Figure 3. Block diagram of the fu PID controller. The fu PID controller signal is given b Eq. (8) (Çetin and Demir, 2008). U PID t ( t) S K U ( i) + K U ( t) = u PI i= 0 PD PD PD (8) The linguistic labels used to describe the fu sets were Negative Big (NB), Negative edium (N), Negative Small (NS), Zero (ZE), Positive Small (PS), Positive edium (P), Positive Big (PB). Inputs and outputs are all normalied in the interval of [-1, 1]. The membership functions are shown in Fig. (4). Figure 4. embership functions. Parameter tuning affects the shape of fu sets. Actual interval of variables is obtained b using scaling factors which are Se, Sde and Su. The adequate selection of the scaling factor contributes to the improvement of control results. The fu control rule is in the form of: IF e = E i and de = de j THAN U PD = U PD(i,j). The table rules used in this paper is similar to that proposed b Çetim and Demir (2008) and is shown in Fig. (5). The amdani tpe is the structure of the rule base. The fu rules are etracted from fundamental knowledge and human eperience about the process. These rules contain the input/output relationships that define the control strateg. Each control input has seven fu sets so that there are at most 49 fu rules. 4 Simulations and Results Figure 4. Rule table of the fu PID controller. Some considerations were done for the development of this work: the satellite is a rigid bod; internal torques are null; reaction wheel friction is null; initial moment is null; constant disturbing torques of N.m in each ais; and a diagonal inertia matri. The reaction wheel parameters are shown in Tab. (1). Workshop on Space Engineering and Technolog, June

5 Table 1. Reaction wheel parameters. Parameters Values Time constant Wheel gain I R [kg.m 2 ] Ω Rma Rma V Rma 7500 [rpm] 0.6 [N.m] 10 [V] T v = 20 [s] K v = 0.06 The simulation of an attitude maneuver of 30 degrees in the roll ais was performed. The initial condition is the following: 30 in the roll ais (φ); 0 in the pitch ais (θ); and 0 in the aw ais (ψ). The mission requirements are: the maneuver must be performed in a maimum time of 180 seconds and reach an appointment accurac of The simulation was performed varing the scaling factor of the error change between 1 and 2, with increment of 0.1. The results of the attitude maneuver simulations in each ais (angular position as function of time) are reported in Figure (5). Figure 5. Angular position as function of time: (a) roll angle; (b) pitch angle; (c) aw angle. As it can be seen from Fig. (5), different kind of response can be achieved varing the scaling factor of the error change (Sde). In order to improve the visualiation, Fig. (6a) shows the pointing accurac in the roll ais as function of Sde and Fig. (6b) shows the settling time as function of Sde. Figure 6. ission requirements: (a) pointing accurac Sde; (b) settling time Sde. In Fig. (6a), it can be seen that a higher precision is obtained for higher values of Sde. The same tendenc of decreasing can be seen in Fig. (6b), but less pronounced. Looking to results presented up to now, we can sa that to attend the specifications, the value of Sde must be equal or higher than 1.3. Workshop on Space Engineering and Technolog, June

6 The angular velocit of the satellite as function of time is presented in Fig. (7). The angular velocit in pitch ais, Fig. (7b), reaches a value different of ero because of the orbital angular velocit considered in the modeling. According to Fig. (7a), the maimum overshoot increases as soon as the Sde increases. Figure 7. Satellite angular velocit as function of time: (a) roll ais; (b) pitch ais; (c) aw ais. The control signal send b fu PID controller is defined as the input voltage to be applied in the reaction wheel. The behavior of this variable is reported in Fig. (8). Looking to Fig. (8a), we can see that the time of voltage saturation varies with the value of Sde. Figure 8. Input voltage as function of time: (a) roll ais; (b) pitch ais; (c) aw ais. Figure (9) shows the time of voltage saturation, in the roll ais, as function of Sde. We can see that how higher the Sde, higher will be the time of voltage saturation. Workshop on Space Engineering and Technolog, June

7 Figure 9. Time of voltage saturation, in the roll ais, as function of Sde. The angular velocit of the reaction wheel, in each ais, is presented in Fig. (10). According to Fig. (10a), the reaction wheel reached a maimum angular velocit of almost 2800 rpm for a Sde equals to 1, and a maimum angular velocit of approimatel 4000 rpm with a Sde equals to 2. Figure 10. Reaction wheel angular velocit as function of time: (a) roll ais; (b) pitch ais; (c) aw ais. The torque applied b reaction wheel in each ais is presented in Fig. (11). Figure 11. Torque applied b reaction wheel as function of time: (a) roll ais; (b) pitch ais; (c) aw ais. Workshop on Space Engineering and Technolog, June

8 Based on the results presented in this paper, the value of 1.3 was chosen for the scaling factor of the change of error (Sde) as the best parameter to be used in the design of the fu PID controller. This value represents a minimum settling time (96.9 s), a minimum time of voltage saturation (10.7 s), and a maimum pointing accurac (0,01 ), considering the mission requirements presented before. 5 Conclusions This paper presented an attitude maneuver of a satellite equipped with reaction wheel and fu PID controller. The performance of the controller has been analed for a range of values of Sde. The design of the fu PID controller was carried out according to some mission requirements. The stud allowed to choose the best value of Sde considering the following requirements: maimum pointing accurac, minimum settling time and minimum time of voltage saturation. The influence of the scaling factor variation in the satellite attitude, like angular position and velocit, also were reported. The use of fu PID controller can be an interesting alternative in design of attitude control sstem. However, it is necessar to carr out an investigation, analsis, modeling and simulation for the correct design of the control sstem, mainl in the case of comple sstems, like artificial satellites. References Lee, C. C. Fu logic in control sstems: fu logic controller, Part I&II. IEEE Transactions on Sstems, an, and Cbernetics. Vol. 20, No. 02, pp , ar-apr Nguen, H. T., Prasad, N. R., Walker, C. L., Walker, E. A., A first course in fu and neural control. 1. ed. Boca Raton, Florida, USA, Chapman & Hall/CRC, 2003, 305 p. ISBN( ). Sidi,. J., Spacecraft dnamics and control: a practical engineering approach. 1. ed. New York, USA: Cambridge Universit Press, p. ISBN( ). Soua,. L. O., Estudo e desenvolvimento de um sistema de controle de atitude ativo em três eios para satélites artificiais usando atuadores pneumáticos a gás frio e volantes de reação. 368 p. (INPE TDL/042). Dissertação (estrado em Ciência Espacial) Instituto Nacional de Pesquisas Espaciais (INPE), São José dos Campos, Soua, P. N. De., Análise, projeto, construção e testes de um modelo de roda de reação para aplicações espaciais. 185 p. (INPE-4358-TDL/299). Dissertação (estrado em Ciência Espacial/ecânica Orbital) Instituto Nacional de Pesquisas Espaciais (INPE), São José dos Campos, Tewari, A., Atmospheric and space flight dnamics: modeling and simulation with ATLAB and Simulink. 1. ed. New York, USA: Birkhäuser Boston, p. ISBN( ). Çetim, S.; Demir, Ö., Fu PID controller with coupled rules for a nonlinear quarter car model. World Academ of Science, Engineering and Technolog. Vol. 41, pp , Workshop on Space Engineering and Technolog, June